Mechanical Engineering

Section 1:
Linear Algebra
Matrix
Algebra Systems of linear equations, eigenvalues and eigenvectors.
Calculus
Functions of single variable, Limit, Continuity and differentiability, Mean value theorems, Indeterminate forms, Evaluation of definite and improper integrals, Double and triple integrals, Partial derivatives, Total derivative, Taylor Series (in one and two variables), Maxima and minima, Fourier series, Gradient, divergence and curl, Vector identities, Directional derivatives, Line, surface and volume integrals, Applications of Gauss, Stokes and Greens theorems.
Differential equations
First order equations (linear and non-linear) higher order linear differential equations with constant coefficients, Euler-Cauchy equation, Initial and boundary value problems, Laplace transforms, Solutions of heat, wave and Laplace's equations.
Complex variables
Analytic functions, Cauchy-Riemann equations, Cauchy's integral theorem and integral formula, and Taylor and Laurent series.
Probability and Statistics
Definitions of probability, sampling theorems, conditional probabilityMean, median, mode and standard deviation; random variables, binomial, Poisson and normal distributions.
Numerical Methods
Numerical solutions of linear and non-linear algebraic equations, Integration by trapezoidal and Simpsons rules, and Single and multi-step methods for differential equations.

You will get mechanical branch in VIT, Vellore but in second or third category (i. e. you have to pay more for that desired branch). Let me explain you the admission system here. Rank in VIT to get admission should be 90,000, so you can get admission in VIT from the 6th category as well. There are various categories as well. With each category admission, fees go high. First category closed till 15,000 rank, and you can get Mechanical till 16000 rank but there are 2nd category as well, if you want mechanical, but you have to pay more for that. Additional Information: Predict engineering colleges you can get on the basis of your JEE Rank: https://www.shiksha.com/b-tech/resources/jee-mains-college-predictor?utm_source=cconnect&utm_medium=CR_JEECP&utm_campaign=8321427.

Dear Sayshay,
Use the link below to check the college
https://www.shiksha.com/b-tech/resources/jee-mains-college-predictor
With this rank, it will get somewhat difficult for you to get some good government college. Instead of settling down for some mediocre government college, it would be better if your try for some good private university.
There are many reputed private universities like Lovely, BITS, VIT, SRM, etc. That are offering B.Tech degree with various specializations. They have their different criteria for admissions. Some offer admission on basis of JEE scores, some offer admission on basis of their own tests and some offer admission on the basis of both. So, if you have low rank in JEE, you can try for admission on basis of your marks in class 12th or give the university specific test for admission. Some of the universities like LPU also offer scholarships on basis of your performance in class 12 or JEE or even during the course. You can check the details of any of these reputed universities. All these universities have wonderful infrastructure and world-class facilities to offer to students apart from good placements that they offer.
Good luck.